For years the graphic designer Stephanie Specht and I have wanted to collaborate on a project related to the architecture of Louis Kahn (1901-1974). This year, I was invited to a panel discussion on the work and design philosophy of Anne Tyng (1920-2011). Anne was an architect by training and had a deep interest in geometry (and in particular platonic solids). Anne collaborated with Louis Khan on his early seminal projects such as the Trenton Bath House (1955, Trenton, New Jersey, USA), the Yale Art Gallery (1953, New Haven, Connecticut,USA) and the City Tower. Stephanie and I were spontaneously attracted to those three projects because they intersect with our research and design practice.
I would like to start the geometric ceiling found in the Yale Art Gallery because it served as inspiration for Stephanie’s graphic. Khan and his team explored new ways with light in this project. Rather than using lots of glass, they thought about how you can control the incoming light explored screens, meshes and panels for daylighting and a geometric tetrahedral grid ceiling with conceiled artificial lighting. This ceiling is Anne’s work.
Anne used an interlocking geometric pattern of tetrahedra, one of the five platonic solid, arranged in a structure to create this light-filled space. The tetrahedral units are attached to the floor slab. This ceiling is bot elegant and efficient: it allows for a large span interior space, unhindered by columns. The tetrahedral geometry allowed the installation of temporary panels for exhibitions. The tetrahedrons holds the electrical ducts and allow the light to diffuse. About the ceiling Kahn said “it is beautiful and it serves as an electric plug.” 
In 1953, three years after the completion of the Yale Art Gallery, Anne travelled to Italy and worked with the structural designer Pier Luigi Nervi (1891-1974). In 1951, Nervi completed the spectacular floor slabs of the Gatti Wool Factory (Lazio, Italy). At first sight, there is a clear visual similarity between the polyhedral ceiling of the Yale Art Gallery and the ribbed floor slabs of the Gatti Wool Factory. However the ribs in the polyhedral ceiling are arranged rectilinear pattern while the ribs in Nervi’s floor slab curve like the veins in a leaf. The orientation and these ribs is not random. They actually follow the principal stresses lines. While the principal stress line inspiration for the rib patterns of Nervi’s floor systems is well-documented, the method used to generate these patterns is considerably unknown by comparison. The concept of aligning floor ribs along the isostatics of the principal bending moments is attributed to Aldo Arcangeli, a practicing engineer who worked in the office of Soc. Ing. Nervi & Bartoli . Arcangeli determined from classical plate theory that a 2D continuous body subjected to normal forces produces two families of orthogonal curves (isostatics), tangential to the principal bending moment trajectories, along which torsional moments are equal to zero. If this continuous body is replaced by ribs oriented along the isostatics, then the rib structure and the continuous body would have identical structural behavior under identical loading and support conditions . Nervi drew inspiration from the patterns formed by the isostatics; he stated: “it was amazing to find that by thus limiting our task to the interpretation of a purely physical phenomenon, we were able to discover unexpected and expressive new forms ,” and asserted that “in harmony with such inspiration, reinforced concrete beams lose the rigidity of wooden beams or of metal shapes and ask to be molded according to the line of the bending moments” .
We developed a tool based on Finite Element Analysis to generate isostatic ribs patterns in floor slabs and demonstrated a correlation between these rib patterns and the rib patterns in Nervi’s floors. 
The idea of arranging material in an efficient way might have appealed to Anne as she was very much interested in growth patterns in nature. She studied growth patterns, occurring in nature, in her PhD thesis “Simultaneous Randomness and Order: the Fibonacci-Divine Proportion as a Universal Forming Principle.”
In Anne’s time the arrival of the electromagnetic microscope allowed researchers to study structure and geometry at much smaller scale. In this X-Ray we see for example that the boney trabeculae inside the human hipbone are also arranged according to the principal stress trajectories. In particular, we see the mode of action of the spaceframe, a system Anne widely explored, in the configuration of the trabeculae. The weight of the body rests on the rounded know at the end of the “cantilever”. With respect to the axis of the hip bone, we get a bending moment M=Pxa. This moment is not constant. It varies with the forces developed as the body moves. The position of the body and the degree of rotation of the joint affect the direction of the resultant loads . The boney trabeculae is arranged to resist all these variations in loading conditions. These multiple systems are fused into a single three dimensional unit. The bone is able to resist forces coming from any direction. The resemblance between these trabeculae and the spaceframes Anne favored are startling; they are both efficient, resilient and 3D repetitive and allow for many different loading combinations.
Anne pioneered space-frame architecture. In her writings and her many sketches, she shows how combining the tetrahedral units can form planes and volumes to built forms on all scales from domestic to the urban scale. In the early 1950s, Tyng designed an elementary school for Bucks County, Pa., in which the roof was a space frame . The school was never built. In this physical model shown below, shows how horizontal tetrahedral plane ( a roof and ceiling) can merge into one point and become a structural support. It is not clear whether this structure was envisaged in concrete or steel. However a spaceframe might have presented an “over” structured approach for the small span of the classroom.
Anne was fascinated with Platonic solids, three-dimensional, multi-sided volumes like tetrahedrons, cubes, octahedrons, dodecahedrons and icosahedrons and how we might inhabit such geometric forms. She thought that the five Platonic solids were the most basic archetypes that formed all organic micro- and macrocosmic structures. Platonic solids have interesting properties such as their polarity or duality where faces and vertices can be interchanged.
We wonder what Anne’s might have thought of our work. We explore how planar polygons transistion to 3D polyhedra solely based on geometry and mechanics of curved crease folding. Like Anne we use physical models but unlike Anne, we now also can exploit the potential of parametric physics based models.
The transformational geometry in Anne’s work was static and very visible in the City Tower Project. A few months before Tyng had begun designing the City Tower, James Watson and Francis Crick had been credited with Rosalind Franklin’s discovery of the double helix form of DNA. Although the City Tower’s geometry is not based on the double helix, Anne was thinking about static expressions of transformations in architecture.
Anne envisioned the City Tower as a tetrahedral grid floor that shifted and rotated . When Kahn included the City Tower plan in his proposal for a redesign of Midtown Philadelphia, , Philadelphia’s chief urban planner Ed Bacon deemed the proposal too expensive and impractical to be materialized.
Ten years later, construction technology, the scaffolding and spherical nodes were commercially brought to the market. They might have made Anne’s project economically viable.
We would like to conclude with the Trenton Bath House project. Louis Kahn often spoke of the Trenton Bath House, as a turning point in his design philosophy. (This realization is not a bath house and is not in Trenton). ” “I discovered myself after designing that little concrete block bathhouse in Trenton. From this came a generative force which is recognizable in every building which I have done since. “According the book “Louis Kahn to Anne Tyng: The Rome Letters 1953–54”  (Rizzoli), Anne “almost immediately” came up with a plan involving “four symmetrically arranged squares with hipped roofs.” She wrote that the design was inspired by bathhouses she remembered from her childhood in China, where her parents were missionaries. It is said that Bath House was largely Anne’s work: the square rooms, the square columns that are both solid and void at the same time; the voids that hold up the roof; the pyramids both solid and void.
We were attracted by Anne’s work because we discovered that she was a visionary at the intersection of architecture, geometry and scaling principles found in Nature. She pioneered space-frame architecture and invented an entire new formal architectural language before the construction and digital parametric design technologies were available.
Loud, Patricia Cummings and Michael P Mezzatesta. The Art Museums of Louis I. Kahn. Durham, NC: Published by Duke University Press in association with the Duke University Museum of Art, 1989. 52-57.
Nervi, P.L., Structures, Salvadori, G., and Salvadori, M. (trans), F.W. Dodge Corp. (1956).
Iori, T., Le plancher a nervures isostatiques de Nervi, In: Gargiani, R. (ed) L’architrave le plancher la plate-forme: Nouvelle histoire de la construction, Presses polytechniques et universitaires romandes, Lausanne (2012).
Nervi, P.L., Aesthetics and Technology in Building, Harvard University Press (1965).
A.B. Halpern, D. Billington D. and S. Adriaenssens, ‘The ribbed floor slab systems of Pier Luigi Nervi’ Journal of International Association of Shell and Spatial Structures, vol. 54, pp 189-198, 2013.
Anne Griswold Tyng. “”Simultaneous Randomness and Order: the Fibonacci-Divine Proportion as”. repository.upenn.edu. Retrieved May 31, 2019.
Kahn, L. I. (1997). Louis Kahn to Anne Tyng: The Rome Letters, 1953-1954. Rizzoli.